Question
If the silver nanoparticles are each 100 atoms in size, how many silver nanoparticles does the scientist have? How many moles of silver nanoparticles does the scientist have if he has a sample mass of 53.95 grams?
Answers
GPT-4o mini
To determine how many silver nanoparticles the scientist has and how many moles of silver nanoparticles this corresponds to, we first need to calculate the mass of a single silver nanoparticle and then use that information to find out how many nanoparticles are present in the sample mass.
### Step 1: Calculate the mass of a single silver nanoparticle
1. **Atomic mass of silver (Ag)**: Approximately \( 107.87 \, \text{g/mol} \).
2. **Number of atoms in the nanoparticle**: 100 atoms.
3. **Calculate the mass of a single silver atom**:
\[
\text{Mass of one silver atom} = \frac{107.87 \, \text{g/mol}}{6.022 \times 10^{23} \, \text{atoms/mol}} \approx 1.79 \times 10^{-25} \, \text{g/atom}
\]
4. **Calculate the mass of 100 silver atoms**:
\[
\text{Mass of one nanoparticle} = 100 \times 1.79 \times 10^{-25} \, \text{g} \approx 1.79 \times 10^{-23} \, \text{g}
\]
### Step 2: Calculate the number of nanoparticles in the sample
1. **Total mass of the sample**: 53.95 grams.
2. **Calculate the number of nanoparticles**:
\[
\text{Number of nanoparticles} = \frac{\text{Total mass}}{\text{Mass of one nanoparticle}} = \frac{53.95 \, \text{g}}{1.79 \times 10^{-23} \, \text{g}} \approx 3.01 \times 10^{24} \, \text{nanoparticles}
\]
### Step 3: Calculate the number of moles of silver nanoparticles
1. **Moles of silver nanoparticles**:
\[
\text{Moles of nanoparticles} = \frac{\text{Number of nanoparticles}}{6.022 \times 10^{23}} \approx \frac{3.01 \times 10^{24}}{6.022 \times 10^{23}} \approx 5.00 \, \text{moles}
\]
### Summary
- The scientist has approximately \( 3.01 \times 10^{24} \) silver nanoparticles.
- The scientist has approximately \( 5.00 \) moles of silver nanoparticles.
### Step 1: Calculate the mass of a single silver nanoparticle
1. **Atomic mass of silver (Ag)**: Approximately \( 107.87 \, \text{g/mol} \).
2. **Number of atoms in the nanoparticle**: 100 atoms.
3. **Calculate the mass of a single silver atom**:
\[
\text{Mass of one silver atom} = \frac{107.87 \, \text{g/mol}}{6.022 \times 10^{23} \, \text{atoms/mol}} \approx 1.79 \times 10^{-25} \, \text{g/atom}
\]
4. **Calculate the mass of 100 silver atoms**:
\[
\text{Mass of one nanoparticle} = 100 \times 1.79 \times 10^{-25} \, \text{g} \approx 1.79 \times 10^{-23} \, \text{g}
\]
### Step 2: Calculate the number of nanoparticles in the sample
1. **Total mass of the sample**: 53.95 grams.
2. **Calculate the number of nanoparticles**:
\[
\text{Number of nanoparticles} = \frac{\text{Total mass}}{\text{Mass of one nanoparticle}} = \frac{53.95 \, \text{g}}{1.79 \times 10^{-23} \, \text{g}} \approx 3.01 \times 10^{24} \, \text{nanoparticles}
\]
### Step 3: Calculate the number of moles of silver nanoparticles
1. **Moles of silver nanoparticles**:
\[
\text{Moles of nanoparticles} = \frac{\text{Number of nanoparticles}}{6.022 \times 10^{23}} \approx \frac{3.01 \times 10^{24}}{6.022 \times 10^{23}} \approx 5.00 \, \text{moles}
\]
### Summary
- The scientist has approximately \( 3.01 \times 10^{24} \) silver nanoparticles.
- The scientist has approximately \( 5.00 \) moles of silver nanoparticles.